Analytic geometry associates to each point in the Euclidean plane an ordered pair. The red ellipse is associated with the set of all pairs (x,y) such that x2/4 + y2 = 1.
In mathematics, an ordered pair (a, b) is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. (In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.)
Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.)
The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects). For example, the ordered triple (a,b,c) can be defined as (a, (b,c)), i.e., as one pair nested in another.
In the ordered pair (a, b), the object a is called the first entry, and the object b the second entry of the pair. Alternatively, the objects are called the first and second components, the first and second coordinates, or the left and right projections of the ordered pair.
Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs, cf. picture.
mathematics, an orderedpair (a, b) is a pair of objects. The order in which the objects appear in the pair is significant: the orderedpair (a, b) is different...
singleton and an orderedpair, respectively. Tuple may be formally defined from orderedpairs by recurrence by starting from orderedpairs; indeed, an n-tuple...
reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an orderedpair P = ( X , ≤ ) {\displaystyle P=(X,\leq )} of a set...
in P {\displaystyle P} or P − {\displaystyle P^{-}} , for example the orderedpair ( 0 , 0 ) {\displaystyle (0,0)} . Then the integers are defined to be...
Cartesian product of two sets A and B, denoted A × B, is the set of all orderedpairs (a, b) where a is in A and b is in B. In terms of set-builder notation...
{\displaystyle x\cup x^{c}=V} . OrderedPair: For each a {\displaystyle a} , b {\displaystyle b} , the orderedpair of a {\displaystyle a} and b {\displaystyle...
by pairs of real numbers x and y. The order of the components x and y is significant, so such a pair is also called an orderedpair. Such a pair is written...
an orderedpair (a, b) has a as its first element and b as its second element, which means (a, b) ≠ (b, a). While the two elements of an orderedpair (a...
ordinate or y coordinate of the point. For example, if (x, y) is an orderedpair in the Cartesian plane, then the first coordinate in the plane (x) is...
<360^{\circ }} . Vectors can be specified using either orderedpair notation (a subset of ordered set notation using only two components), or matrix notation...
that, two orderedpairs are equal if and only if their first elements are equal and their second elements are equal. Formally, an orderedpair with first...
different contexts. The idempotents of B are all pairs (x, x), where x is any natural number (using the orderedpair characterisation of B). Since these commute...
to the beginning of a list.) Although cons cells can be used to hold orderedpairs of data, they are more commonly used to construct more complex compound...
in the chain. Thus a singleton set is a chain of length zero, and an orderedpair is a chain of length one. The dimension of a space is often defined or...
In one restricted but very common sense of the term, a graph is an orderedpair G = ( V , E ) {\displaystyle G=(V,E)} comprising: V {\displaystyle V}...
given field, commonly the real numbers), and such that for any given orderedpair of points there is a unique translation sending the first point to the...
Cartesian plane. The set R 2 {\displaystyle \mathbb {R} ^{2}} of the orderedpairs of real numbers (the real coordinate plane), equipped with the dot product...
addressed Ordered set, an ordered structure, in mathematics Ordinate in mathematics, the y element of an orderedpair (x, y) Partially ordered set Complete...
regularity enables defining the orderedpair (a,b) as {a,{a,b}}; see orderedpair for specifics. This definition eliminates one pair of braces from the canonical...
something, a pair Unordered pair, or pair set, in mathematics and set theory Orderedpair, or 2-tuple, in mathematics and set theory Pairing, in mathematics...
In mathematics, a fundamental pair of periods is an orderedpair of complex numbers that defines a lattice in the complex plane. This type of lattice is...
orderedpair (a pair ( x , y ) {\displaystyle (x,y)} which is the same type as its projections) in NFU. It is convenient to use the Kuratowski pair in...
is called a proportion. Consequently, a ratio may be considered as an orderedpair of numbers, a fraction with the first number in the numerator and the...
relation as an orderedpair using the null set. At approximately the same time, Hausdorff (1914, p. 32) gave the definition of the orderedpair (a, b) as {{a...
2nd edition, p. 9. A. Kanamori, "The Empty Set, the Singleton, and the OrderedPair", p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21...
Global Information